Combined interior and boundary <elliptic regularity> for the Dirichlet Laplacian on a smooth bounded domain states that, for every integer $k\geq0$,
$$
\|u\|_{H^{k+2}(\Omega)}
\leq C_k\|f\|_{H^k(\Omega)}
$$
when $\Delta u=f$ and $u$ has zero boundary trace. More generally one first has an additional $\|u\|_{L^2}$ term, which uniqueness and the Poincare inequality remove here.

For the shifted equation, write $\Delta u=f+u$. The weak estimate gives $u\in H^1$. Applying the displayed estimate first with an $L^2$ right side gives $u\in H^2$. Repeating,
$$
f\in H^k,\quad u\in H^j
\quad\Longrightarrow\quad
u\in H^{\min(k,j)+2},
$$
until $u\in H^{k+2}$. The lower-order term is controlled at each stage, yielding
$$
\|u\|_{H^{k+2}}\leq C_k\|f\|_{H^k}.
$$
This is <boundary elliptic regularity for the shifted Dirichlet Laplacian>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 2 / d / Solution

On a smooth bounded domain, a solution of
$$
(\Delta-c)u=f,\qquad u|_{\partial\Omega}=0,
$$
for an invertible shift satisfies
$$
\|u\|_{H^{k+2}}\leq C_k\|f\|_{H^k}.
$$
Interior regularity and boundary flattening give the derivative gain, while invertibility controls the lower-order norm.


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